Let $V,W$ be nonzero spaces over a field $F$ and suppose that a set $B =\lbrace v_1, . . . , v_n \rbrace \subset V$ has the following property:
For any vectors $w_1, . . . ,w_n \in W$, there exists a unique linear transformation $T : V \rightarrow W$ such that $T(v_i) = w_i$ for all $i = 1, . . . , n$.
Prove that $B$ is a basis of the space $V$.
Can anyone help me with this? I have no idea how to prove this.